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A survey of alternating permutations. (English) Zbl 1231.05288
Brualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4/pbk). Contemporary Mathematics 531, 165-196 (2010).
Summary: A permutation \(a_1a_2\cdots a_n\) of \(1,2,\dots, n\) is alternating if \(a_1> a_2< a_3> a_4<\cdots\) . We survey some aspects of the theory of alternating permutations, beginning with the famous result of D. AndrĂ© [C. R. 88, 965–967 (1879; JFM 11.0187.01)] that if \(E_n\) is the number of alternating permutations of \(1,2,\dots, n\), then \(\sum_{n\geq 0} E_n{x^n\over n!}= \text{sec\,}x+ \tan x\). Topics include refinements and \(q\)-analogues of \(E_n\), various occurrences of \(E_n\) in mathematics, longest alternating subsequences of permutations, umbral enumeration of special classes of alternating permutations, and the connection between alternating permutations and the \(cd\)-index of the symmetric group.
For the entire collection see [Zbl 1202.05003].

MSC:
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05A05 Permutations, words, matrices
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
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